Exercise 10, § 8

Let X be a compact topological space and A a unital Banach subalgebra of $\mathscr{C}(X)$ for the induced norm.

a) A subset K of X is said to be antisymmetric (relative to A) if every function $f\in A$ that is real on K is constant on K. Every antisymmetric subset is contained in a maximal antisymmetric subset. A maximal antisymmetric subset is closed. The set $\mathfrak{R}$ of maximal antisymmetric subsets is a partition of X.

b) Let $A^{\bot}$ be the orthogonal of A in the Banach space of complex measures on X. Let $\mu$ be an extreme point of the unit ball of $A^{\bot}$. Now prove that the support of $\mu$ is antisymmetric.

c) Let $f\in \mathscr{C}(X)$. If $f|K\in A|K$ for every $K\in \mathfrak{R}$, then $f\in A$. (Apply b) and the Krein–Milman theorem.) Hence recover the Weierstrass–Stone theorem (TG, X, p. 36, th. 3).

d) A nonempty subset E of X is called a peak (relative to A) if there exists $f\in A$ such that $\|f\|= 1$ and E is the set of $x\in X$ for which $f(x) = 1$. One may then suppose that $|f(x)|<1$ for $x\notin E$ (replace $f$ by $\frac{1}{2}(1 +f)$). Every nonempty countable intersection of peaks is a peak. If $E_1,E_2$ are peaks, then $E_1\cup E_2$ is a peak. (Let $f_1, f_2\in A$ with $f_i= 1$ on $E_i,|f_i|<1$ on X - $E_i$; let $g_i=\frac{1}{4}(1-f_i)^{1/3}\in A$; then $1-g_1g_2= 1$ on $E_1\cup E_2,|1-g_1g_2|<1$ on X - $(E_1\cup E_2)$.)

e) Let E be an intersection of peaks. Let I be the set of $f\in A$ vanishing on E. Then $f\mapsto f|E$ defines an isometry of $A/I$ onto the algebra $A|E$ of restrictions to E of functions in A.

f) Let E be a peak. Let $E'\subset E$ be a peak relative to $A|E$. Then $E'$ is a peak relative to A.

g) Every antisymmetric subset of X is an intersection of peaks. (Use d) and f).) h) For every antisymmetric subset K of X, the algebra $A|K$ is a closed subalgebra of $\mathscr{C}(K)$. (Use e) and g).)

i) Let R be the set of real parts of the elements of A. Suppose that $X\in \mathfrak{R}$ and that R is stable under multiplication. Then X is reduced to a point. (Let $x_0\in X$. For $u\in R$, there exists a unique $f\in A$ such that $u=\mathscr{R}(f)$ and $f(x_0)\in \mathbf{R}$; set $N(u) =\|f\|$. Then R is a real Banach space for N; the closed graph theorem proves that multiplication in R is separately continuous, hence continuous. Deduce on $S = R +iR$ a Banach algebra structure. Let $p\in R$ with $p(x)>0$ for every $x\in X$. Now prove that, for every character $\chi$ of S, one has $\mathscr{R}(\chi (p))>0$, and deduce that log($p$)$\in R$. Assuming $X\not=\{x_0\}$, one can construct an $f\in A$ such that $0<\mathscr{R}(f)\leqslant 1$ on X$,f(x_0)\in \mathbf{R}$ and such that $\|f\|$ is arbitrarily large. From the above, there exists $V\in A$ with $|V|^2=\mathscr{R}(f)$. One has $\|V\|\leqslant 1$ therefore $N(\mathscr{R}(V))\leqslant 2$, but

1

2

is arbitrarily large.)

j) If R is stable under multiplication, then $A =\mathscr{C}(X)$. (Use a), c), h), i).) Consequently, if $A\not=\mathscr{C}(X)$, there exists $u\in R$ such that $u^2\notin R$.